Probability and Visualization

OIS Chapter 3, Jaynes, Grammar of Graphics

Robert W. Walker

2025-09-16

Overview

AMA
Basic Data Visualization: The FAQ
Introducing esquisse
The Economist: Mistakes
Tables
Probability

AMA

Data

Code

load(url("https://github.com/robertwwalker/DADMStuff/raw/master/Week3Data.RData"))

Basic Data Visualization

Introducing Esquisse

Esquisse

Executing

esquisse:::esquisser(viewer="browser")

NB: It needs to run in a separate browser window.

Selecting Data

What is available in the environment?

A Run Through

Probability and Tables

Do’s and Dont’s

The Economist: Mistakes

Probability

Two rules:

Probabilities sum to one.
The probability of any event is greater than or equal to zero.

Where does Probability Come From?

There are three common sources of probabilities:

Known formula [Dice, Coins, etc.]
Empirical frequency
Subjective belief

A priori probability

The probability of a given integer on a k-sided die: \frac{1}{k}.

The probability of heads with a fair coin: \frac{1}{2}.

The probability of a Queen? \frac{4}{52}

The probability of a Diamond? \frac{13}{52}

The Queen of Diamonds? \frac{1}{52} or (\frac{4}{52}\times\frac{13}{52})

Quasirandom numbers

Empirical probability: frequency

How often does something happen?

Annie

Straight to Watch

This is Historical Statistics

How likely am I to be admitted? Consult the admissions rate

How fast do I drive? Likelihood of law enforcement and need for speed

In data: this is tables.

Berkeley

        
           No  Yes
  Female 1278  557
  Male   1493 1198

    M.F   No  Yes
 Female 1278  557
   Male 1493 1198

Code

library(tidyverse)
library(janitor)
table(UCBAdmit$M.F,UCBAdmit$Admit)
UCBAdmit %>% tabyl(M.F, Admit)

Three Versions

Code

prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit), 1)

        
                No       Yes
  Female 0.6964578 0.3035422
  Male   0.5548123 0.4451877

Code

prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit), 2)

        
                No       Yes
  Female 0.4612053 0.3173789
  Male   0.5387947 0.6826211

Code

prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit))

        
                No       Yes
  Female 0.2823685 0.1230667
  Male   0.3298719 0.2646929

Code

UCBAdmit %>% tabyl(M.F, Admit) %>% adorn_percentages("row")

    M.F        No       Yes
 Female 0.6964578 0.3035422
   Male 0.5548123 0.4451877

Code

UCBAdmit %>% tabyl(M.F, Admit) %>% adorn_percentages("col")

    M.F        No       Yes
 Female 0.4612053 0.3173789
   Male 0.5387947 0.6826211

Code

UCBAdmit %>% tabyl(M.F, Admit) %>% adorn_percentages("all")

    M.F        No       Yes
 Female 0.2823685 0.1230667
   Male 0.3298719 0.2646929

Plot It

Code

( UCBM <- ggplot(UCBAdmit) + aes(x=M.F, fill=Admit) + geom_bar(position="dodge") + scale_fill_viridis_d() )

Subjective Probability

How likely do we believe something is?

The Great Divide

Empirical frequency vs. subjective belief

Empirical Frequency: She’s Right

Physics Disagrees: We Goin Nova…..

Annie’s a liar.

What matters in group decision making is probably as much the beliefs [subjective] as the evidence [frequency].

How should we reflect this in strategies of argumentation/persuasion?

Think

What matters?

Code

# RUN ME
# may need to install.packages("countdown")
library(countdown)
countdown_fullscreen(
  minutes = 5, seconds = 0,
  margin = "5%",
  font_size = "8em",
)

05:00

Three Concepts from Set Theory

Intersection [and]
Union [or] avoid double counting the intersection
Complement [not]

Three Distinct Probabilities

Joint: Pr(x=x and y=y)
Marginal: Pr(x=x) or Pr(y=y)
Conditional: Pr(x=x | y=y) or Pr(y =y | x = x)

Joint Probability

The table sums to one.

For Berkeley:

Code

UCBAdmit %>% tabyl(M.F, Admit) %>% adorn_percentages("all")

    M.F        No       Yes
 Female 0.2823685 0.1230667
   Male 0.3298719 0.2646929

Code

prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit))

        
                No       Yes
  Female 0.2823685 0.1230667
  Male   0.3298719 0.2646929

Marginal Probability

The row/column sums to one. We collapse the table to a single margin. Here, two can be identified. The probability of Admit and the probability of M.F.

Code

UCBAdmit %>% tabyl(M.F)

    M.F    n   percent
 Female 1835 0.4054353
   Male 2691 0.5945647

Code

UCBAdmit %>% tabyl(Admit)

 Admit    n   percent
    No 2771 0.6122404
   Yes 1755 0.3877596

Code

prop.table(table(UCBAdmit$M.F))


   Female      Male 
0.4054353 0.5945647

Code

prop.table(table(UCBAdmit$Admit))


       No       Yes 
0.6122404 0.3877596

Conditional Probability

How does one margin of the table break down given values of another? Each row or column sums to one

Four can be identified, the probability of admission/rejection for Male, for Female; the probability of male or female for admits/rejects.

For Berkeley:

Code

UCBAdmit %>% tabyl(M.F, Admit) %>% adorn_percentages("row")

    M.F        No       Yes
 Female 0.6964578 0.3035422
   Male 0.5548123 0.4451877

Code

prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit), 1)

        
                No       Yes
  Female 0.6964578 0.3035422
  Male   0.5548123 0.4451877

Code

UCBAdmit %>% tabyl(M.F, Admit) %>% adorn_percentages("col")

    M.F        No       Yes
 Female 0.4612053 0.3173789
   Male 0.5387947 0.6826211

Code

prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit), 2)

        
                No       Yes
  Female 0.4612053 0.3173789
  Male   0.5387947 0.6826211

Law of Total Probability

Is a combination of the distributive property of multiplication and the fact that probabilities sum to one.

For example, the probability of Admitted and Male is the probability of admission for males times the probability of male.

Pr(x=x, y=y) = Pr(y | x)Pr(x)

Or it is the probability of being admitted times the probabilty of being male among admits.

Pr(x=x, y=y) = Pr(x | y)Pr(y)

Now the Substance

The ggplot fill aesthetic is great for displaying these things. For example, are males and females equally likely to be admitted to Berkeley?

Plaintiffs say no.

Code

ggplot(UCBAdmit) + aes(x=M.F, fill=Admit) + geom_bar() + scale_fill_viridis_d()

Is that an Adequate Comparison?

The University says no. Why? The most important factor in the probability of admission is likely to be the department. This has a huge impact on what we see.

Code

ggplot(UCBAdmit) + 
  aes(x=M.F, fill=Admit) + 
  geom_bar(position="fill") + 
  scale_fill_viridis_d() + 
  facet_wrap(vars(Dept))

The Magic of Bayes Rule

To find the joint probability [the intersection] of x and y, we can use either of the aforementioned methods. To turn this into a conditional probability, we simply take it is a proportion of the relevant margin.

Pr(x | y) = \frac{Pr(y | x) Pr(x)}{Pr(y)}